14
votes
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Calculating alpha and its meaning
Alphas from a time-series regression are error terms in the cross-sectional, linear relationship between expected returns and factor betas. If a factor model were correct those error terms (the alphas)...
- 6,924
12
votes
Accepted
Which algorithms do robo-advisors use?
After having done a lot of research on the topic I found the following excellent research piece on ETF.com:
Wealthfront modifies historic asset-class returns with current market
implied expected ...
- 27.3k
12
votes
Accepted
Fama-French factor model: why mimicking portfolios?
The innovation of Fama and French's Three Factor Model wasn't in finding book to market ratios forecast returns but in reconciling that empirical regularity with the standard framework of macro-...
- 6,924
9
votes
Accepted
Portfolio construction in reality?
Portfolio optimalisation depends heavily on the estimation of the moments (and therefore has HUGE estimation uncertainty).
Even though it's useful for comparing and analysing different existing ...
- 333
8
votes
Accepted
Why do anomalies disappear after they get detected?
The best explanation I have seen so far is the so-called Adaptive Market Hypothesis by Andrew Lo:
The adaptive market hypothesis, as proposed by Andrew Lo, is an
attempt to reconcile economic ...
- 27.3k
8
votes
Accepted
How are modern portfolio theory (MPT) and CAPM related?
CAPM states that the expected return of any given asset should equal $ER_i=R_f+β_i (R_m-R_f)$, with α being the error term of the previous equation. Now, as α has an expected value of zero, then only ...
- 516
8
votes
Closed-form analytical solution for the variance of the minimum-variance portfolio?
A few more steps beyond your last equation gives the answer.
With $C = \mathbf{1}^T\mathbf{\Sigma}^{-1}\mathbf{1}$, we have
$$\sigma_P^2 = [C^{-1} \mathbf{\Sigma}^{-1}\mathbf{1}]^T \mathbf{\Sigma} [C^{...
- 3,595
7
votes
Accepted
Marginal Risk Contribution Formula
concerning your first question: the derivative does not disappear: $\sigma(R_p)$ contains the square root.
To be more precise, set
$$
\sigma(R_p) = \sqrt{w_1^2\cdot\sigma(R_1)^2 + w_2^2\cdot\sigma(R_2)...
- 1,436
7
votes
What’s the derivative of the sharpe ratio for one asset? Trying to optimize on it for a model
I agree that the paper could be much clearer: what it calls the “Sharp ratio derivative” is actually the “differential Sharpe ratio” proposed in a NIPS paper by Moody & Safell.
In Section 2.2 of ...
- 200
7
votes
Most significant research articles for practical investors with research perspectives
A lot has happened since Markowitz and Sharpe. While their work is still considered foundational, the empirical/practical relevance of their models has been questioned by later work.
Here are a few ...
- 9,332
7
votes
Portfolio Optimization and Global Minimum Variance Portfolio (GMV)
1) To be honest, any horizon is problematic in this respect. Simple sampling statistics 101 will tell you that the standard error around any estimate of true mean returns is the root time * variance. ...
- 5,041
7
votes
Contribution of an asset's variance to portfolio variance
In this answer, I am assuming that you want to keep correlations constant.
To begin with, note that the $N\times N$ covariance matrix $\Sigma$ with element $\Sigma_{i,j}=Cov(x_i,x_j)$ can be written ...
- 6,425
6
votes
Accepted
Mean Variance Portfolio theory and real-world problem?
Mean-variance (MV) is a framework rather than a prescription. This framework allows one to make, discuss, and defend his investment decision.
In practice, there are many ways to make adjustments to ...
- 751
6
votes
Accepted
Maximum Certainty Equivalent Portfolio with Transaction Costs
Seems like a small mistake in the last equation. It should read
$\Delta^* = A^{-1} \left[\mu-\gamma \Sigma \omega_c - \frac{1}{\iota'A^{-1}\iota} \iota' A^{-1}(\mu-\gamma \Sigma \omega_c )\iota\...
- 116
6
votes
Accepted
Generalized Mean Variance Portfolio
Something to perhaps realize is that your two problems may not be as different as you think if $\lambda$ is an ad-hoc parameter.
For any solution to your 2nd problem (where $\theta > 1$), there ...
- 6,924
6
votes
Accepted
Rockafellar-Uryasev mean-CVaR optimiztion
$VaR_\alpha$ is a scalar choice variable in the minimization problem. In the Rockafeller-Uryasev paper, it is simply called $\alpha\in R$. (C.f., the program described in Theorem 2 of that paper, or ...
- 405
6
votes
Accepted
Portfolio Optimization sum of weights constraint with short selling
In the early days of Portfolio Theory there were different views about short positions. Some authors modeled short positions as negative and required all weights to add up to 1 (first equation), ...
- 10.9k
6
votes
Accepted
Why isn't the asset with minimum variance given a 100% portfolio weight?
Diversification is key.
The clear cut answer is diversification. A weighted combination of assets will more often than not show a lower return variance than even the asset with the lowest variance ...
- 6,425
5
votes
Basic question on Portfolio Theory
Of course estimating expected returns is the very core of portfolio management. Finding a useful covariance matrix too. To find both fills a book. So I first thought about closing the question. But it ...
- 13.6k
5
votes
Which algorithms do robo-advisors use?
Well, I did some modest research on this topic, looking at peers.
Most of them use Modern Portfolio Theory, see this pic:
You can find this small survey here: https://www.linkedin.com/pulse/...
- 51
5
votes
Accepted
Portfolio with lots of subportfolios
One way to this is the following (you can code all these constraints if you use the right software, I am doing such things using mathematica)
You define $w_{i,j}$ which is the weight of asset $j$ in ...
- 13.6k
5
votes
Accepted
Interpretation of portfolio standard deviation
It depends on the distribution of the returns. If you assume that it's roughly normally distributed, then you have a ~68% chance for a return in the range of 1 standard deviation, ~95% chance for 2 ...
- 788
5
votes
Accepted
how can we know the residual return will be uncorrelated with the market return
Let us ignore the riskless rate for simplicity of the presentation.
If you have (historical or simulated) return series $r_i$ for the portfolio and $r_i^M$ for the market, then the beta is the OLS ...
- 13.6k
5
votes
Accepted
Why did Markowitz not derive an equation for the efficient frontier?
It is surprising. What I think is: Markowitz became interested in the general problem when there are constraints (including inequality constraints) on the portfolio weights (in addition to the ...
- 9,332
5
votes
In portfolio theory, has volatility a logical place as an asset class?
IMO: Volatility is a risk factor not an asset class. Asset classes are collections of assets and volatility is not one. Options, volatility derivatives, etc, are asset classes which might offer ...
- 4,307
5
votes
Backtest Results needed to Model Validate my Modern Portfolio Theory model
There is a recent a paper recently using a population test of all CRSP data from 1925-2013 as a test of whether a mean and a variance exist versus they do not exist. It overwhelmingly excluded mean-...
- 4,359
5
votes
Accepted
Optimisation with strong correlated Assets
This answer will try and outline all the different possibilities I came across over the last couple of years, including drawbacks. But first, let me outline the problem a little.
To appreciate the ...
- 2,915
5
votes
Why do anomalies disappear after they get detected?
Any anomaly that can be phrased as a "mispricing" or "relative value" opportunity can be expected to disappear as more people discover it and trade on it.
For example, say that stock movements over ...
- 5,891
5
votes
Accepted
Finding a minimum variance portfolio when using a regulariser?
You're not going to get an analytic formula except in special cases of function $\rho(x)$. And you're probably going to want $\rho$ convex.
If $\rho$ is convex, the problem is a convex optimization ...
- 6,924
5
votes
Accepted
Widely accepted methods for coming up with the co-variance matrix of assets?
Multivariate volatility models for replacing the sample covariance matrix with in the mean-variance portfolio selection model:
RiskMetrics 1996 EWMA (Exponentially weighted moving average) covariance ...
- 2,980
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